CONSERVATION LAWS ON THE SPHERE: FROM SHALLOW WATER TO BURGERS - PowerPoint PPT Presentation

CONSERVATION LAWS ON THE SPHERE: FROM SHALLOW WATER TO BURGERS Matania Ben-Artzi Institute of Mathematics, Hebrew University, Jerusalem, Israel Advances in Applied Mathematics IN MEMORIAM OF PROFESSOR SAUL ABARBANEL Tel Aviv University December 2018 joint work with JOSEPH FALCOVITZ, PHILIPPE LEFLOCH 1

“...together with David Gottlieb we noticed that some of the stuff that people were doing, the formulation was not strongly well posed, which is a mathematical point of view. So we got interested in how to make it more posed.” (Interview with P. Davis, Brown University,2003). “Problems should be studied in a ‘physico-mathematical’ fashion”–(private communication) 2

General Circulation Model –JETSTREAM 3

General Circulation Model –JETSTREAM 4

MOVING VORTEX R.D. Nair and C. Jablonowski–Moving vortices on the sphere: A test case for horizontal advection problems, Monthly Weather Review 136(2008)699–711 5

✥ ✑ ✏ ✑ ☎ � ✒ ☎ ✒ ✍ ✏ ✓ ✓ ☎ ✏ ✔ ✎ ☎ ✌ ✝ ✁ ☞ ✄ ✂ ☎ ✆ ✂ ✝ ✞ ✞ ✟ ✠ ✡ ☛ -1 -.8 -.6 -.4 -.2 0 .2 .4 .6 .8 1 6

Grid on sphere—the Kurihara Grid D. J. Williamson–The evolution of dynamical cores for global atmospheric models, Journal of the meteorological society of Japan 85B (2007)241–269 DISCUSSION: The “POLE PROBLEM” φ λ 7

SOME REFERENCES–GEOPHYSICAL P.S. Marcus–Numerical simulation of Jupiter’s great red spot , Nature (1988) 8

SOME REFERENCES–GEOPHYSICAL P.S. Marcus–Numerical simulation of Jupiter’s great red spot , Nature (1988) J.Y-K. Cho, and L. M. Polvani–The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets , Science (1996) 9

SOME REFERENCES–GEOPHYSICAL P.S. Marcus–Numerical simulation of Jupiter’s great red spot , Nature (1988) J.Y-K. Cho, and L. M. Polvani–The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets , Science (1996) J. Galewski, R.K. Scott and L. M. Polvani–An initial-value problem for testing numerical models of the global shallow water equations , Tellus (2004) 10

SOME REFERENCES–GEOPHYSICAL P.S. Marcus–Numerical simulation of Jupiter’s great red spot , Nature (1988) J.Y-K. Cho, and L. M. Polvani–The morphogenesis of bands and zonal winds in the atmospheres on the giant outer planets , Science (1996) J. Galewski, R.K. Scott and L. M. Polvani–An initial-value problem for testing numerical models of the global shallow water equations , Tellus (2004) T. Woollings and M. Blackburn–The North Atlantic jet stream under climate change and its relation to the NAO and EA patterns , (NAO=North Atlantic Oscillations,EA=East Atlantic) Journal of Climate (2012) 11

SOME REFERENCES–COMPUTATIONAL D. L. Williamson, J.B. Drake, J.J. Hack, R. Jakob and P. N. Swarztrauber–A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comp. Physics (1992) 12

SOME REFERENCES–COMPUTATIONAL D. L. Williamson, J.B. Drake, J.J. Hack, R. Jakob and P. N. Swarztrauber–A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comp. Physics (1992) J. A. Rossmanith, D. S. Bale and R. J. LeVeque–A wave propagation method for hyperbolic systems on curved manifolds , J. Comp. Physics (2004) 13

SOME REFERENCES–COMPUTATIONAL D. L. Williamson, J.B. Drake, J.J. Hack, R. Jakob and P. N. Swarztrauber–A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comp. Physics (1992) J. A. Rossmanith, D. S. Bale and R. J. LeVeque–A wave propagation method for hyperbolic systems on curved manifolds , J. Comp. Physics (2004) P. A. Ullrich, C. Jablonowski and B. van Leer–High-order finite-volume methods for the shallow water equations on the sphere , J. Comp. Physics (2010) 14

SOME REFERENCES–COMPUTATIONAL D. L. Williamson, J.B. Drake, J.J. Hack, R. Jakob and P. N. Swarztrauber–A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comp. Physics (1992) J. A. Rossmanith, D. S. Bale and R. J. LeVeque–A wave propagation method for hyperbolic systems on curved manifolds , J. Comp. Physics (2004) P. A. Ullrich, C. Jablonowski and B. van Leer–High-order finite-volume methods for the shallow water equations on the sphere , J. Comp. Physics (2010) L. Bao, R.D. Nair and H.M. Tufo–A mass and momentum flux-form high-order discontinuous Galerkin shallow water model on the cubed-sphere , J. Comp. Physics (2013) 15

SOME BOOKS D. R. Durran–Numerical Methods for Fluid Dynamics: With Applications to Geophysics , Springer (1999,2010) N. Paldor–Shallow Water Waves on the Rotating Earth , Springer (2015) R. Salmon–Lectures on Geophysical Fluid Dynamics, Oxford University Press (1998) 16

GRP METHODOLOGY USING RIEMANN INVARIANTS and GEOMETRIC COMPATIBILITY J. Li and G. Chen–The generalized Riemann problem method for the shallow water equations with topography, Int. J. Numer. Methods in Engineering(2006) M. Ben-Artzi, J. Li and G. Warnecke–A direct Eulerian GRP scheme for compressible fluid flows, J. Comp. Physics (2006) M. Ben-Artzi, J. Falcovitz and Ph. LeFloch– Hyperbolic conservation laws on the sphere: A geometry compatible finite volume scheme, J.Comp. Physics (2009) 17

DERIVATION OF THE MODEL INVARIANT FORM 18

TWO SYSTEMS Lower-case letters = Inertial system Capital letters =Rotating system . q ( t ) , ˙ Time derivatives of vector functions: ˙ � � Q ( t ) . Connection by ROTATION MATRIX x = R ( t ) � X . � x = d ˙ dt R ( t ) � X = R ( t )( � Ω × � � X ) . � Ω = � Ω( t ) = angular velocity in the rotating system. It is constant (namely, independent of time) in the rotating system. 19

If � X ( t ) represents a moving particle in the rotating system, x = d X ) + R ( t )( ˙ ˙ dt R ( t ) � X = R ( t )( � Ω × � � X ) . � 20

If � X ( t ) represents a moving particle in the rotating system, x = d X ) + R ( t )( ˙ ˙ dt R ( t ) � X = R ( t )( � Ω × � � X ) . � Ω × ˙ Ω × ˙ X + ¨ ¨ x = R ( t ) { � Ω × ( � Ω × � X ) + � X + � � � � � X } Ω × ˙ X + ¨ = R ( t ) { � Ω × ( � Ω × � X ) + 2 � � � X } . 21

If � X ( t ) represents a moving particle in the rotating system, x = d X ) + R ( t )( ˙ ˙ dt R ( t ) � X = R ( t )( � Ω × � � X ) . � Ω × ˙ Ω × ˙ X + ¨ ¨ x = R ( t ) { � Ω × ( � Ω × � X ) + � X + � � � � � X } Ω × ˙ X + ¨ = R ( t ) { � Ω × ( � Ω × � X ) + 2 � � � X } . Particle of mass m , force � f (in the inertial system): R ( t )( m ¨ Ω × ˙ � X ) = � f − mR ( t ) { � Ω × ( � Ω × � X ) + 2 � � X } . Lagrangian formulation: particle has unit mass , and is an element of a fluid continuum moving (approximately) on the spherical surface of the earth S . . � N = outward unit normal on the sphere S . 22

TWO “ADDITIONAL FORCES” Ω × ( � � Ω × � CENTRIFUGAL FORCE X ) Ω × ˙ 2 � � CORIOLIS FORCE X V = ˙ � � Velocity X . 23

ASSUMPTION I: There are two body forces acting on the particle: ◮ − � G — the gravity force . ◮ � H — the hydrostatic force (due to fluid pressure). Total force (in the inertial system) on the unit mass is � f = R ( t )( − � G + � H ) . 24

ASSUMPTION I: There are two body forces acting on the particle: ◮ − � G — the gravity force . ◮ � H — the hydrostatic force (due to fluid pressure). Total force (in the inertial system) on the unit mass is � f = R ( t )( − � G + � H ) . R ( t )( ¨ Ω × ˙ X ) = R ( t ) {− � � G + � H − � Ω × ( � Ω × � X ) − 2 � � X } . 25

ASSUMPTION I: There are two body forces acting on the particle: ◮ − � G — the gravity force . ◮ � H — the hydrostatic force (due to fluid pressure). Total force (in the inertial system) on the unit mass is � f = R ( t )( − � G + � H ) . R ( t )( ¨ Ω × ˙ X ) = R ( t ) {− � � G + � H − � Ω × ( � Ω × � X ) − 2 � � X } . Note: � X is a three-dimensional vector in the rotational system. Later: Confine to the sphere S : r = a , by assuming that the fluid volume is very “thin”( vertically). 26

ASSUMPTION I: There are two body forces acting on the particle: ◮ − � G — the gravity force . ◮ � H — the hydrostatic force (due to fluid pressure). Total force (in the inertial system) on the unit mass is � f = R ( t )( − � G + � H ) . R ( t )( ¨ Ω × ˙ X ) = R ( t ) {− � � G + � H − � Ω × ( � Ω × � X ) − 2 � � X } . Note: � X is a three-dimensional vector in the rotational system. Later: Confine to the sphere S : r = a , by assuming that the fluid volume is very “thin”( vertically). ¨ Ω × ˙ X = − � � G + � H − � Ω × ( � Ω × � X ) − 2 � � X . 27

ASSUMPTION II: Some constant g ∗ > 0 , X ) = − g ∗ � − � G − � Ω × ( � Ω × � N . 28